Of course, there are too many significant things a successful project manager must pay attention to them. However, as per my experience, the following issues do matter the most from my perspective:
1- A good PM must be proactive and should try to eliminate negative things before they happen rather than fixing them.
2- Does he/she put the project interest at the top and work for it or not?, for example, a good PM must not be distracted by interests of stakeholders or personal relations or gaining personal benefits.
3- Communication with/managing people(stakeholders). See, modern project management is about managing people mostly and we all know that this is one of the hardest management fields (managing people). managing cost, time, risks...etc can be managed, there are enormous tools to help with that, But people, this is the challenge. I STRONGLY BELIEVE that a successful project manager must be able to wear different hats with each different type of person.